Pair Correlation of Zeros and Primes in Short Intervals

In 1943, A. Selberg [15] Deduced From The Riemann Hypothesis (Rh) that $$\int\limits_{\rm{1}}^{\rm{X}} {{{\left( {\psi \left( {\left( {{\rm{1 + }}\delta } \right){\rm{x}}} \right){\rm{ - }}\psi \left( {\rm{x}} \right){\rm{ - }}\delta {\rm{x}}} \right)}^2}{{\rm{x}}^{{\rm{ - 2}}}}{\rm{dx}} \ll \delta {{\left( {{\mathop{\rm l}\nolimits} {\rm{ogX}}} \right)}^2}}$$ (1) for X–1 ≤ δ ≤ X–1/4, X ≥ 2. Selberg was concerned with small values of δ and the constraint δ ≤ X–1/4 was imposed more for convenience than out of necessity. For Larger δ we have the following result.